Irregular network

ABSTRACT

Irregularities are provided in at least one dimension of a torus or mesh network for lower average path length and lower maximum channel load while increasing tolerance for omitted end-around connections. In preferred embodiments, all nodes supported on each backplane are connected in a single cycle which includes nodes on opposite sides of lower dimension tori. The cycles in adjacent backplanes hop different numbers of nodes.

BACKGROUND OF THE INVENTION

An interconnection network consists of a set of nodes connected bychannels. Such networks are used to transport data packets betweennodes. They are used, for example, in multicomputers, multiprocessors,and network switches and routers. In multicomputers, they carry messagesbetween processing nodes. In multiprocessors, they carry memory requestsfrom processing nodes to memory nodes and responses in the reversedirection. In network switches and routers, they carry packets frominput line cards to output line cards. For example, pending U.S. patentapplication Ser. No. 08/918,556, filed by William J. Dally, Philip P.Carvey, Larry R. Dennison and P. Alan King for Internet Switch Router,describes the use of a three-dimensional torus interconnection networkto provide the switching fabric for an internet router.

The arrangement of the nodes and channels in a network defines itstopology. FIG. 1, for example, illustrates a two-dimensional, radix-8torus topology, also called an 8-ary 2-cube. In this topology each nodeis designated by a two-digit radix-8 address. Channels in bothdirections connect each pair of nodes whose addresses differ by one inexactly one digit modulo 8. That is, node (a,b) is connected to nodes(a−1,b),(a+1,b),(a,b−1), and (a,b+1) where the addition and subtractionare performed modulo-8. Each node in this topology has degree 4 since itis connected to four other nodes. A related topology, an 8-ary 2-mesh,is shown in FIG. 2. This network is identical to the 8-ary 2-cube exceptthat the ‘end-around’ connections from node 7 to node 0 in eachdimension are omitted. While the mesh network is simpler to wire, it hassignificantly lower performance than the torus network as will bedescribed below.

Meshes and tori may be constructed with different radices and numbers ofdimensions, and other network topologies are possible such as crossbars,trees, and butterfly networks. Several popular topologies are described,for example, by William J. Dally, “Network and Processor Architecturesfor Message-Driven Computers,” in VLSI and Parallel Computation, Editedby Suaya and Birtwistle, Morgan Kaufmann, 1990.

The topology chosen for an interconnection network is constrained by thetechnology used to implement the network. Pin-count constraints onchips, circuit boards, and backplanes, for example, limit the maximumdegree of each network node. Wire-length constraints limit the maximumdistance a single channel can traverse when the network is mapped into aphysical packaging structure.

Low-dimensional (2-3 dimensions) torus and mesh networks have becomepopular in recent years in part because they offer good performancewhile having a small node degree and uniformly short wires when mappedto two or three dimensional electronic packaging systems. While at firstit may appear that the end-around channels in a torus network result inlong wires, these long wires can be avoided by folding the torus asillustrated in FIG. 3. The figure shows how an 8×4 torus that is foldedin the x-dimension is mapped onto a set of backplanes 20. Each node ispackaged on a circuit card and eight of these circuit cards are insertedin each backplane 20. The circuit cards in each backplane span two xcoordinates and the entire y dimension. The y-dimension channels arerealized entirely in the backplane while the x-dimension channels arerealized by connections between backplanes. Folding the torus allowsthis topology to be realized with each x-channel connecting no furtherthan the adjacent backplane.

To expand the network of FIG. 3, it is necessary to first break theloopback connection 22 or 24 at the end of the x-dimension and then makea connection to a new backplane. An efficient method for reconfiguringthe network in this manner is disclosed in pending U.S. patentapplication Ser. No. 09/083,722, filed by Philip P. Carvey, William J.Daily and Larry R. Dennison for Apparatus and Methods for ConnectingModules Using Remote Switching.

Two important measures of the quality of a topology are average pathlength and maximum channel load. The average path length, D_(avg), of anetwork is the average number of channels that must be traversed to senda packet between two randomly selected nodes in a network. In theradix-8, two-dimensional torus network of FIG. 1, the average packettraverses two channels in each dimension, giving P_(avg)=4. In general,the average path length in a k-ary n-cube is kn/4 (for k even) andn(k/4-1/4k) (for k odd) while the average path length in a k-ary n-meshis n(k/3-1/3k). Average path length is a good predictor of networklatency. The lower the average path length of a topology, the lower isthe latency of a network using that topology, assuming the latency of anindividual channel remains constant.

The maximum channel load of a network, C_(max), is the largest number ofpackets that traverse a single channel when all nodes send packets toall other nodes divided by the number of nodes squared. In the 8-ary2-cube network of FIG. 1, for example, the channel load is uniform withC_(max)=C_(i)=1 for every channel i. In general, the maximum channelload in a k-ary n-cube is k/8 while the maximum channel load in a k-aryn-mesh is k/4. Maximum channel load is a good predictor of thethroughput of a network. The lower the maximum channel load of atopology, the higher is the throughput of a network using that topology,assuming the bandwidth of an individual channel remains constant.

SUMMARY OF THE INVENTION

While the folded torus network offers a method of packaging a regulark-ary n-cube network so that all of the channels remain short, it isinherently inefficient because nodes that are packaged very close to oneanother can communicate only via circuitous routes. For example, in FIG.3, nodes (2,0) and (7,0) are on the same backplane. A message from (2,0)to (7,0), however, must leave this backplane and traverse fivebackplane-backplane connections before arriving back at the backplane.The problem becomes even more acute as the x-dimension grows in radix. Amore efficient topology would minimize this off-backplane communicationby allowing direct communication between nodes on the same backplane. Amore efficient topology would also provide good performance withoutneeding an end-around connection that must be removed for expansion.

One mechanism for increasing system performance in terms of average pathlength and maximum channel load and for making a network lesssusceptible to loss of end around connections is to add one or moredimensions to the network. However, adding a dimension increases thenode degree by two, thus increasing the complexity of each node.

In accordance with one aspect of the invention, benefits of an addeddimension without the increased node degree can be obtained byconnecting one subset of channels within the network in one dimensionand connecting another subset of the network channels in a differentdimension. By distributing the nodes connected by the two channelsubsets throughout the network, messages transferred through the networkalways have ready access to each dimension, but since each node needonly be connected to one dimension or the other, there is no increase indegree. The concept of interleaving differing connections along one ormore dimensions can be extended to other configurations for improvedperformance. Further, connections can be modified to assure that allnodes on a backplane, for example, are connected in a single cycle. Theincreased path lengths between some nodes resulting from the addition ofsuch local connections can generally be masked by alternating theconfigurations from backplane to backplane.

In accordance with the invention, a plurality of channels connect aplurality of nodes in each connection network. A first subset of thechannels connects the nodes is in a first dimension. A second subset ofthe channels connects the nodes in at least a second dimension, andconnection of the second subset of channels differs at differentcoordinates along the first dimension.

In a preferred embodiment, the second subset of the channels connectsthe nodes in cycles or linear arrays in a second dimension and theordering of the nodes in cycles or linear arrays is different atdifferent coordinates along the first dimension. A connection of thesecond subset of channels at a first set of coordinates in the firstdimension connects nodes whose coordinates in the second dimensiondiffer by a first number. The connection of the second subset ofchannels in a second set of coordinates in the first dimension connectsnodes whose coordinates in the second dimension differ by a secondnumber. Thus, for example, in even numbered positions along thex-dimension, adjacent nodes in the backplane may be connected. However,in odd numbered positions, every other node may be connected.

Preferably, cycles or linear arrays of the second subset of the channelsconnect multiple nodes of each of plural cycles or linear arrays in thefirst dimension. For example, where the channels of the first dimensionare folded, the second subset of the channels may connect nodes onopposite sides of the folds, thus reducing loading in the firstdimension.

A first set of coordinates in the first dimension of the second subsetof channels may connect the nodes into a cycle or linear array along asecond dimension while, at a second set of coordinates in the firstdimension, the second subset of channels connect the nodes into a cycleor linear array along a third dimension.

A preferred embodiment of the invention is a three-dimensional array.The second subset of the channels includes channel connecting nodes in asecond dimension and channels connecting nodes in a third dimension. Theconnections in the third dimension differ at different coordinates ofthe first and second dimensions.

Nodes connected in at least the second dimension at a coordinate alongthe first dimension may be packaged in a single packaging module, suchas on a backplane or other circuit board.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features and advantages of theinvention will be apparent from the following more particulardescription of preferred embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingthe principles of the invention.

FIG. 1 is an illustration of a prior art 8-ary 2 cube.

FIG. 2 is an illustration of a 8-ary 2 mesh.

FIG. 3 is an illustration of a prior art folded torus packaged onto fivebackplanes.

FIG. 4 is an alternating 3 cube embodying the present invention packagedonto five backplanes.

FIG. 5 is the alternating 3 cube of FIG. 4 with alternate node labelingshowing the 3 cube nature of the network.

FIG. 6 is an illustration of an interleaved cube topology embodying thepresent invention.

FIGS. 7A and 7B illustrate the y-dimension connections for theinterleaved cube of FIG. 6 in the even and odd backplanes, respectively.

FIG. 8 illustrates a prior art 20-node backplane for a doubly folded 3cube.

FIG. 9 illustrates one z plane of a prior art doubly folded torusshowing connection between four backplanes.

FIG. 10 illustrates an even numbered backplane for a three-dimensionalinterleaved torus embodying the invention.

FIG. 11 shows the z connections for the even numbered backplane in thethree-dimensional interleaved torus.

FIG. 12 illustrates the z connections for an odd numbered backplane inthe three-dimensional interleaved torus.

FIG. 13 illustrates the z connections for an even numbered backplane inan alternative three-dimensional interleaved torus.

FIG. 14 illustrates the z connections for an even numbered backplane ina second alternative to the three-dimensional interleaved torus.

DETAILED DESCRIPTION OF THE INVENTION

A network that allows more local communication between nodes that arephysically close to one another is shown in FIG. 4. In this networkeven-numbered backplanes E are wired identically to the backplanes ofFIG. 3 while odd-numbered backplanes O connect the nodes on oppositesides of each x-loop that shares the backplane. This arrangement allowsa node to access any other node on its backplane in at most six hops andcrossing at most two backplane connectors.

The topology of FIG. 4 is best thought of as a three-dimensionalnetwork. The x-dimension selects the backplane, the y-dimension selectsa pair of nodes within the backplane (as before), and the z-dimensionselects the node within the pair. FIG. 5 shows the same network with thenodes renumbered to correspond to this three-dimensional interpretationof the network. Every node in this network has connections to itsneighbors in the x-direction while only ‘even’ nodes have connections inthe y-direction and only ‘odd’ nodes have connections in thez-direction. Because the connections alternate between y- and z- werefer to this network as an alternating cube. This particularalternating cube has a radix-5 mesh in the x-dimension, a radix-4 torusin the y-dimension, and a radix-2 torus in the z-dimension. While theend-around connections form a 10-cycle in the x-dimension, we ignorethose connections for purposes of analyzing this topology and considerthe 10-cycle as two linear arrays of 5-nodes at different coordinates inthe z-dimension. One skilled in the art will understand that thistopology can be generalized to any number of dimensions with any radixin each dimension and either mesh or torus end connections in eachdimension.

The alternating 3-cube of FIGS. 4 and 5 has two advantages compared tothe folded 2-cube of FIG. 3 and one significant disadvantage. First, ithas a slightly shorter average path length (3.26 vs. 3.5). This reflectsthe utility of the connections on the odd backplanes. More importantly,this network works nearly as well with one or both of the end-aroundconnections in the x-dimension omitted. When just one of the connectionsis omitted, for example, the average path length increases slightly from3.26 to 3.30 and the maximum channel load remains constant. In contrast,the average path length of the torus network increases from 3.5 to 4.3and the maximum channel load is doubled (from 1.25 to 2.5) when thisconnection is opened. Thus, a network with an alternating cube topologycan omit the costly hardware required to reconfigure the rightend-around channel to allow expansion. With a torus network, on theother hand, this hardware is strictly required.

The disadvantage of the alternating cube is its throughput. The maximumchannel loading of the network of FIG. 4 is 1.33 vs. 1.25 for the torusof FIG. 3. This 6% increase in maximum channel load corresponds to a 6%decrease in peak network throughput.

The increased C_(max) of the alternating cube network is caused by aload imbalance on channels in the x-dimension. This imbalance can becorrected by connecting the even- and odd-numbered backplanes in aninterleaved cube topology as shown in FIG. 6. Because each node on abackplane can be reached by another node on the backplane withoutcrossing to another backplane, loading of x-channels is reduced. Thisconfiguration has both a lower average path length (2.87 vs. 3.5) and alower maximum channel load (1.20 vs. 1.25) than the torus network. Likethe alternating cube, this network has nearly the same performance(D_(avg)=2.93, C_(max)=1.20) when one of the end-around connections isomitted to save the cost of reconfiguration.

In the topology of FIG. 6, the nodes in each even-numbered backplane arewired in a single cycle that we use to define the y-dimension of thenetwork, rather than the two cycles of FIG. 5. All nodes in FIG. 6 arelabeled with a two-digit address where the first digit gives thebackplane number (x-coordinate) and the second digit gives they-coordinate. Moving in the x-dimension changes only the x-coordinatewhile moving in the y-dimension changes only the y-coordinate. The nodesin each odd-numbered backplane of FIG. 6 are also connected in a singlecycle, but not the same cycle as the even-numbered backplanes.

The relation between the even-numbered and odd-numbered backplanes inthe interleaved cube is illustrated in FIGS. 7A and B. In theeven-numbered backplanes of FIG. 7A, each node is connected to the nextnode ahead and the next node behind along the cycle. Nodes in theodd-numbered backplanes of FIG. 7B, on the other hand, are connected tothe nodes three hops ahead and behind along the cycle.

It is significant that the even and odd backplanes support differentconfigurations. The single cycle of each backplane enablescommunications between any two nodes on a backplane without leaving thebackplane. However, the single cycle does result in increased distancesbetween some nodes. For example, when node 001 is connected directly tonode 031 in FIG. 5, the corresponding node 04 in FIG. 6 is now threehops away from node 07. However, if one considers a message travelingacross many backplanes in a large system, a message may be routed fromthe lower torus to the upper torus in a single hop from the node 14 tothe node 17. Whether a message is routed to a different x-dimensiontorus in either an even or an odd backplane is determined by which typeof backplane provides the shortest path.

Up to this point we have described the alternating cube and interleavedcube topologies in two dimensions for simplicity. One skilled in the artwill understand that these topologies can be realized with any number ofdimensions.

For example, in a preferred embodiment of the invention, the interleavedcube topology is applied to a 3-dimensional doubly-folded torus network(a doubly-folded 3-cube). A single backplane of the unmodified torusnetwork, shown in FIG. 8, contains 20 nodes in four cycles of five. Eachnode is labeled with a three-digit number giving its relative x, y, andz-coordinates within the backplane. There are six ports on each node.The positive and negative ports in the x-dimension connect to the rightand left edges of the backplane respectively. Similarly the positive andnegative ports in the y-dimension connect to the top and bottom edges ofthe backplane. In contrast, the positive and negative ports in thez-dimension connect each node to the other nodes in its cycle. Thez-dimension is wired entirely within the backplane while the x- andy-dimensions are wired entirely with backplane to backplane connections.

The x- and y-connections between the backplanes of the doubly-folded3-cube network are illustrated in FIG. 9. The figure shows one of thefive z-planes for a 4×4×5 3-cube. The nodes in the other four z-planesare wired identically. The figure illustrates how the network is foldedin both the x- and y-dimensions to realize a torus connection in bothdimensions using only connections to nodes on adjacent backplanes. Notethat since FIG. 9 has been extended to four backplanes, the numbering ofthe nodes does not directly correspond to that of FIG. 8. However, ifone considers the backplane of FIG. 8 to be the upper left backplane ofFIG. 9, the nodes 00z, 03z, 30z and 33z of FIG. 9 correspond, forexample, to nodes 000, 010, 100 and 110 in one z plane or node 001, 011,101 and 111 in another z plane.

FIG. 9 can be compared to two backplanes of FIG. 3. The differencebetween the circuits is that, in the y-dimension, the nodes of FIG. 9are folded in the same manner as in the x-dimension. This allows forshorter end around channels and expansion in the y-dimension throughedge connection of backplanes.

While FIG. 9 shows a 4×4×5 torus, one skilled in the art will understandhow the network can be expanded at the right edge and the top edge toproduce a network with any even number of nodes in the x- andy-dimensions and 5-nodes in the z-dimension. In the preferred embodimentthe network may be expanded to 560-nodes with a 14×8×5 torus topology.

The doubly-folded 3-cube network is converted to a 3-dimensionalinterleaved cube by connecting the even-numbered backplanes asillustrated in FIGS. 10 and 11 and the odd-numbered backplanes asillustrated in FIG. 12. In the doubly-folded torus network, aneven-numbered backplane is one in which the sum of the x- andy-coordinates of the backplane is even. The even- and odd-numberedbackplanes alternate in a checkerboard manner as the network isexpanded.

As shown in FIG. 10, the nodes of each even-numbered backplane areconnected in a single 20-cycle. FIG. 11 shows this same 20-cycle butwith the x- and y-dimension channels omitted for clarity. Advancingalong the z-channels of the network moves a packet through the 5-nodesin the z-dimension, but then, rather than looping back as in FIG. 8, thenext hop in the z-dimension jumps to another side of an x- or y-torus.Referring to FIG. 9, the single cycle of FIG. 10 would connect node 00znot only to the other four nodes in its z cycle of FIG. 8 but also tothe other z cycles of FIG. 8. For example, in FIG. 9, node 00z alsoconnects to node 03z on the opposite side of the y-torus fold, node 30zon the opposite side of the x-torus fold and to node 33z on completelydifferent x- and y-tori. As described above in the case of the 2-Dinterleaved cube network, this connection facilitates communicationbetween nodes that are physically close to one another even though theymay be at opposite positions along the x- or y-torus network.

The connections between the nodes on the odd-numbered backplanes for the3-dimensional interleaved torus are shown in FIG. 12. As with theeven-numbered backplanes, the nodes are connected in a 20-cycle.However, it is a different 20-cycle than in the even-numberedbackplanes. As seen by a comparison of FIGS. 11 and 12, while advancingalong a z-channel in an even-numbered backplane (FIG. 11) moves one stepforward or backward along the z-cycle, advancing over a z-channel in anodd-numbered backplane (FIG. 12) moves three steps forward or backwardalong the cycle.

Somewhat better performance in channel loading can be achieved byconnecting the even backplanes of the interleaved 3-D torus network intwo 10-cycles as shown in FIG. 13. With this connection, advancing alonga z-channel moves two steps forward or backward along the z-cycle.However, because the step size, two, is not relatively prime to thelength of the cycle, 20, the connection does not form a single 20-cycle,but rather forms two 10-cycles. For applications where it is importantthat all of the nodes be connected in a single 20-cycle in thez-dimension, this topology can be modified as illustrated in FIG. 14. Inthis figure the two 10-cycles of FIG. 13 have been fused by connectingnodes 004 and 010 and nodes 003 and 011. The irregular topology of FIG.14 has slightly lower performance than the topology of FIG. 13 but hasthe advantage that a single backplane is completely connected withoutany external end-around connections. However, the irregularitycomplicates the computation of routes.

One skilled in the art will understand that there are many variations onthe interleaved topology described here. In particular there may be anynumber of dimensions (not just 2 or 3) and each dimension may beorganized as a torus or a mesh or even a more complex topology such as abutterfly or tree. The network need not be packaged on backplanes butmay use some other packaging technology and hierarchy. For example, thenodes may be directly cabled together. The last dimension of the networkneed not be strictly local to a single backplane. The last dimension ofthe network need not be organized into a single cycle, but rather couldbe arranged as a group of cycles or linear arrays, or some combinationof the two. Also, the cycles in the odd-numbered backplanes need notalways advance by three relative to the even numbered backplanes. Theycan advance by any number that is relatively prime to the radix of thecycle and can even advance by irregular amounts.

While this invention has been particularly shown and described withreferences to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the spirit and scope of theinvention as defined by the appended claims.

What is claimed is:
 1. An interconnection network comprising: aplurality of nodes; and a plurality of channels connecting the nodes: afirst subset of the channels connecting the nodes in a first dimension;and a second subset of the channels connecting the nodes in cycles orlinear arrays in a second dimension, the ordering of the nodes in thecycles or linear arrays being different at different coordinates alongthe first dimension; the connection of the second subset of channels ata first set of even-numbered coordinates in the first dimensionconnecting nodes whose coordinates in the second dimension differ by afirst number; and the connection of the second subset of channels at asecond set of odd-numbered coordinates in the first dimension connectingnodes whose coordinates in the second dimension differ by a secondnumber.
 2. An interconnection network as claimed in claim 1 wherein; ata first set of coordinates in the first dimension the second subset ofchannels connect the nodes into a cycle or linear array along a seconddimension; and at a second set of coordinates in the first dimension thesecond subset of channels connect the nodes into a cycle or linear arrayalong a third dimension.
 3. An interconnection network as claimed inclaim 1 wherein the network connects the nodes of a parallel computer.4. An interconnection network as claimed in claim 1 wherein the networkconnects line cards in a network router.
 5. An interconnection networkas claimed in claim 1 wherein the nodes connected in at least the seconddimension at a coordinate along the first dimension are packaged in asingle packaging module.
 6. An interconnection network as claimed inclaim 5 wherein all nodes in the packaging module are connected in asingle cycle within the module.
 7. An interconnection network as claimedin claim 5 wherein the single packaging module is defined by a circuitboard.
 8. An interconnection network as claimed in 5 wherein the singlepackaging module is defined by a backplane.
 9. An interconnectionnetwork as claimed in claim 1 wherein the first subset of channels formscycles or linear arrays that are folded.
 10. An interconnection networkas claimed in claim 1 wherein all nodes in a packaging module areconnected in a single cycle within the module.
 11. An interconnectionnetwork as claimed in claim 1 wherein the second subset of the channelsincludes channels connecting nodes in a second dimension and channelsconnecting nodes in a third dimension, the connections in the thirddimensions differing at different coordinates of the first and seconddimensions.
 12. An interconnection network as claimed in claim 1 whereinchannels of the second subset form cycles, each of which intersects eacharray of the first subset of channels at multiple nodes.
 13. Aninterconnection network as claimed in claim 1 wherein each of theplurality of channels directly connects a distinct pair of the nodes.14. An interconnection network comprising: a plurality of nodes; and aplurality of channels connecting the nodes: a first subset of thechannels connecting the nodes in a first dimension; and a second subsetof the channels connecting the nodes in cycles or linear arrays in asecond dimension, the ordering of the nodes in the cycles or lineararrays being different at different coordinates along the firstdimension, the cycles or linear arrays of the second subset of thechannels connecting multiple nodes of each of plural cycles or lineararrays in the first dimension, the first subset of the channels beingcycles or linear arrays that are folded and the second subset of thechannels connecting nodes on opposite sides of the folds.
 15. Aninterconnection network as claimed in claimed 14 wherein each of theplurality of channels directly connects a distinct pair of the nodes.16. An interconnection network comprising a plurality of nodes; and aplurality of channels connecting the nodes: a first subset of thechannels connecting the nodes in a first dimension; a second subset ofthe channels connecting the nodes in a second dimension; a third subsetof the channels connects the nodes in cycles or linear arrays in a thirddimension, the ordering of the nodes in the cycles or linear arraysbeing different at different coordinates along the first and seconddimensions; the connection of the third subset of channels at a firstset of coordinates in the first and second dimensions connecting nodeswhose coordinates in the third dimension differ by a first number; theconnection of the third subset of channels at a second set ofcoordinates in the first and second dimensions connecting nodes whosecoordinates in the third dimension differ by a second number; and thefirst set of coordinates corresponding to locations where the sum of thecoordinates in the first and second dimensions is even.
 17. Aninterconnection network as claimed in claim 16 wherein each of theplurality of channels directly connects a distinct pair of the nodes.18. A method of connecting nodes in the network comprising: connectingthe nodes in a first dimension with a first subset of channels; andconnecting the nodes in at least a second dimension with a second subsetof channels, the second subset of channels connecting the nodes incycles or linear arrays in a second dimension, the ordering of the nodesin the cycles or linear arrays being different at different coordinatesalong the first dimension; the connection of the second subset ofchannels at a first set of coordinates in the first dimensioncorresponding to even-numbered positions connecting nodes whosecoordinates in the second dimension differ by a first number; and theconnection of the second subset of channels at a second set ofcoordinates in the first dimension corresponding to odd-numberedpositions, connecting nodes whose coordinates in the second dimensiondiffer by a second number.
 19. A method as claimed in claim 18 whereinthe first subset of channels forms cycles or linear arrays that arefolded.
 20. A method as claimed in claim 18 wherein each channel of thefirst and second subsets of channels directly connects a distinct pairof the nodes.
 21. A method of connecting nodes in the networkcomprising: connecting the nodes in a first dimension with a firstsubset of channels; and connecting the nodes in at least a seconddimension with a second subset of channels, the second subset ofchannels connecting the nodes in cycles or linear arrays in a seconddimension, the ordering of the nodes in the cycles or linear arraysbeing different at different coordinates along the first dimension; thecycles or linear arrays of the second subset of the channels connectingmultiple nodes of each of plural cycles or linear arrays in the firstdimension; the first subset of the channels being cycles or lineararrays that are folded and the second subset of the channels connectingnodes on opposite sides of the folds.
 22. A method as claimed in claim21 wherein: at a first set of coordinates in the first dimension thesecond subset of channels connect the nodes into a cycle or linear arrayalong a second dimension; and at a second set of coordinates in thefirst dimension the second subset of channels connect the nodes into acycle or linear array along a third dimension.
 23. A method as claimedin claim 21 wherein the network connects the nodes of a parallelcomputer.
 24. A method as claimed in claim 21 wherein the networkconnects line cards in a network router.
 25. A method as claimed inclaim 21 wherein the nodes connected in at least the second dimension ata coordinate along the first dimension are packaged in a singlepackaging module.
 26. A method as claimed in claim 25 wherein all nodesin the packaging module are connected in a single cycle within themodule.
 27. A method as claim 25 wherein the single packaging module isdefined by a circuit board.
 28. A method as claimed in claim 25 whereinthe single packaging module is defined by a backplane.
 29. A method asclaimed in claim 21 wherein all nodes in a packaging module areconnected in a single cycle within the module.
 30. A method as claimedin claimed on 21 wherein the second subset of the channels includeschannels connecting nodes in a second dimension and channels connectingnodes in a third dimension, the connections in the third dimensionsdiffering at different coordinates of the first and second dimensions.31. An interconnection network as claimed in claim 21 wherein channelsof the second subset form cycles, each of which intersects each array ofthe first subset of channels at multiple nodes.
 32. A method as claimedin claim 21 wherein each channel of the first and second subsets ofchannels directly connects a distinct pair of the nodes.
 33. A method ofconnecting nodes in the network comprising: connecting the nodes with afirst subset of channels in a first dimension; connecting the nodes witha second subset of channels in a second dimension; and connecting thenodes with a third subset of channels in at least a third dimension, thethird subset of the channels connecting the nodes in cycles or lineararrays being different at different coordinates along the first andsecond dimensions; the connection of the third subset of channels at afirst set of coordinates in the first and second dimensions connectingnodes whose coordinates in the third dimension differ by a first number;the connection of the third subset of channels at a second set ofcoordinates in the first and second dimensions connecting nodes whosecoordinates in the third dimension differ by a second number; and thefirst set of coordinates corresponding to locations where the sum of thecoordinates in the first and second dimensions is even.
 34. A method asclaimed in claim 33 wherein each of the channels of the first, secondand third subsets of channels directly connects a distinct pair of thenodes.